Rubik's Cubes Confused Us. What About This?

I was surfing on internet when I came across a really interesting question -

How many scrambled Rubik's Cube configurations exist such that it takes exactly 7 moves to solve the cube.

Now while the problems sounds easy from the view that we just need to find the number of ways to shuffle a Rubik's Cube in 7 moves, it will be incredibly hard to prove that there doesn't exist a shorter way to solve the cube.

Can only one provide solution to the problem?

Details and Assumptions

You are supposed to solve any given configuration in fewest moves possible.
Any $90°$ or $180°$ turn counts as a single turn while $-90°$ and $-180°$ counts as a single turn. Meaning, we are using half or semi turn metric.
To familiarize yourself with Rubik's Cube's notations, please check this website - Rubik's Cube Notation:

Rubik's Cube Notations

If you want to familiarize yourself with how Rubik's Cube works, check this applet.

Bonus, can you find the answer for other value in place of 7?

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int moves[]; /* stores tuples of form (min moves required to achieve the configuration, actual configuration) */ Generate all possible configurations which can be achieved by moving exactly 1 side and store them in array moves[]. Generate all possible configurations which can be achieved by moving exactly 2 side. Then check if the configuration already exists in moves[] array. If it does, discard the configuration or else store it. Generate all possible configurations which can be achieved by moving exactly 3 side. Then check if the configuration already exists in moves[] array. If it does, discard the configuration or else store it. And so on.

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Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
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`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

According to this. there are $100,803,036$ different configuration of the rubiks cube that can be solved with $7$ moves.

Here is the table for the $20$ moves. Above that all of the $43,252,003,274,489,856,000$ configurations are solvable.

image

Beakal Tiliksew - 5 years, 11 months ago

In fact, after it was discovered that all Rubik's Cube configurations were solvable with at most $20$ moves, the number $20$ became known as "God's Number".

Daniel Liu - 5 years, 11 months ago

I know this table. In fact this question arose from this table itself. But is there any way to prove the number?

Arulx Z - 5 years, 11 months ago

This is a very interesting question. One way to think of it is a complicated graph theory question where we are being asked to compute the shortest distance between two nodes

Agnishom Chattopadhyay - 5 years, 11 months ago

I agree. When I saw it, I was like "How do people Think of this?"

It would be interesting to have others' opinions here $\ddot\smile$

Mehul Arora - 5 years, 11 months ago

I don't know about number of moves required but I can solve rubiks cube in 50 seconds.

AJ S - 5 years, 11 months ago

My record is 18 secs.

3x3x3 rubik cube has 43.252.003.274.489.856.000 .The above figure is only the state can reach by turning the face. If including the possible status as Rubik's cube disassemble and reassemble the figures up to 519.024.039.293.878.272.000 permutation. But i think that this cube can be solved at most 22 step or less, this information i get from Tomas Rokicki

Phong Huỳnh Phú - 5 years, 11 months ago

It is 20 or less for half turn metric. It was considered to be 22 or less before a recent research.

Arulx Z - 5 years, 9 months ago

Hey How to decrease the solving time, even after a lot of practice I am not able to do that

Sarthak Sharma - 5 years, 8 months ago

Try buying a lubricated speed cube. Learn new techniques like CFOP etc and learn look ahead method.

Arulx Z - 5 years, 8 months ago

It is easy enough to calculate the first few numbers like for 7 : 1st move can be any of the 63=18 moves (counting U2 as 1 move ) 2nd move can be any of 53 moves 3rd move can be another 53 moves(repetitions begin) 4th move 53 moves... 7th move 5*3 moves.

however if 1st move is a U move 2nd is a D move and 3rd is a U move again, then it can be solved in less steps. So such pairs result in there being about 148 such possibilities and thus from the above calculated number we subtract 148*(6 moves required to solve cube) However we also need to remember to remove the repitions of sequences like U,D',U2,D2 so effectively we need to subtract 148 times all the numbers in right side.

The answer which you get from this is pretty close (last 4 digits miss) and very time consuming(You need previous answers to get next answer) So this is not at all efficient method, also once you cross 10 moves you get trickier repititions which can be solved by other shortcuts and that means that not really good for more than 4 or 5.

Ajinkya Shivashankar - 4 years, 8 months ago

This analysis would be very difficult to do. But here's another not so efficient method which may work if optimized.

I'll try to improve the run time. In the meanwhile, you can try modelling the Rubik's cube moves into something like a mathematical group and define associate, multiplicative properties etc. Maybe this will make the search very easy...

Arulx Z - 4 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$